Potentially the sets of insights developed in ancient Chinese
culture, and fundamental to the philosophy of life and governance, is therefore
especially relevant at this time. Great importance was then attached to both
the coherence of the pattern as a whole and its capacity to model the changes
experienced and to be anticipated. The use of three of them for purposes
of divination [review],
now considered inappropriate by some [discussion],
should not distract from their insights into the operations of complex psychosocial
systems.

This is especially the case
in the light of the increasing emphasis now placed on "values" and
"wisdom" in relation to global governance, whether "faith-based" or
not, and on elaboration of more appropriate strategies. If the sets of insights
effectively functioned, or were used, as what would now be termed "global
models" or "world
models", then the adequacy of their predictive capacity should be distinguished
from the adequacy of their descriptive/explanatory capacity -- especially
given the requirements of them by their culture.
The most comprehensive of these present day models of the world system
face a particular challenge in that the "predictions" they
offer fail to engage the individual as coherent and credible -- and therefore
fail to engender political support for appropriate concerted action (cf Donnella
Meadows, et al. Limits
to Growth: the 30-Year Update, 2004).

Experimental context

This is an exercise in building on the tables in the separate
paper 9-fold
Higher Order Patterning of Tao Te Ching Insights: possibilities in the mathematics
of magic squares, cubes and hypercubes (2003). There the 81 insights
of the Tao Te Ching (Book
of the Way and its Virtue) were presented experimentally in cells in
a 9 x 9 square to explore the possible existence of higher order patterns
of significance. Titles have here been added to the insights in the cells
-- but these titles are derived from the 81 insights of the T'ai Hsüan
Ching (Tai
Xuan Jing / Canon of Supreme Mystery /The Great Dark Mystery)
of Yang Hsiung (Yang
Xiong), thus establishing a relationship between the
two sets of insights. This
paper is therefore a development of the previous one only through the addition
of points relating to the T'ai Hsüan Ching in order to faciliate
comprehension of any possible relationship with the Tao Te Ching.

The Mystery was the most influential among the many meant to remedy inconsistencies in the Changes and to add to the old discourse current ideas about the cosmic order, the sagely life, and the beauty and precision that can be drawn from words. Until the thirteenth century Yang Hsiung's
writings were considered central to the orthodox search for universal pattern,
and thereafter were forgotten...The Mystery made considerable demands on
its readers. The clarity of its structure was intentionally balanced by
the complexity of language that strives above all for allusiveness....The Mystery,
like the Changes, was said to be hopelessly abstruse and of no practical
benefit....Yang's aim in writing the Mystery [was] to instigate and guide
the personal striving for integrity that is the only possible basis for
a sound polity. This virtue is more than a matter of moral and psychic
integration; it involves union with the Way of Nature and its Mystery....It
does not offer magical power over nature. It simply aids reflection on
the eternal patterns that underlie every aspect of experience and action.
Assimilating those patterns, Yang was convinced, could guide the renewal
of human creativity and the eventual recovery of order....His book applies rigorously and reflects, in its texts and guides to interpretation, the basic seasonal rhythms, the fundamental social relationships, and the functions of yin-yang and the Five Phases that pervade the natural and human worlds.

Although Nylan and Sirvin make no explicit mention of any relation
between the Mystery and the Tao Te Ching, as purportedly
articulated by Lao-Tzu, they
do, like others, refer to the Tao Te Ching as "the Lao-Tzu"
and note Yang Hsiung's recognition of such a connection in the following terms:

It is from Lao-tzu's Mystery that that of Yang derives, although his moral
stance differs: "As for Lao-tzu's discussion of the Way and its power, I
have drawn upon it; but from his rejection of Good (jen) and Right (i),
his elimination of ritual and study, I have taken nothing."

As with the I
Ching,
the T'ai Hsüan Ching was orginally one of several works
that formed the Ta Pu or
Grand Oracle. It is considered to be a companion volume to the I Ching --
which is far better known. Like the Tao Te Ching, the T'ai Hsüan
Ching has
81 insights known in this case as Shou. Like the I Ching these
are associated with a diagram of broken and unbroken lines. In the case of
the 64 insights of the I Ching, each is represented by six such
lines (a hexagram), each of which may be unbroken (yang), or broken once
only (yin). In
the case of the T'ai
Hsüan
Ching,
these are represented by four such lines (a tetragram or quadgram), each
of which may be unbroken, or broken once or twice. The
sequence of numbers in the Tai Hsuan Ching is conventionally arranged
into three groups of three called T'ien (1-27), Jen (28-54)
and Ti (55-81) as discussed below.

Each of the 64 hexagrams of the I Ching is traditionally
associated with a descriptive name for the explanatory details associated
with it. There is no such descriptive name associated with the 81 insights
of the Tao
Te Ching , which are each presented through a set of poetic verses.
These were reduced, experimentally, to a single phrase in an earlier phase
of this experiment (Tao
Te Ching Interpreted Succinctly: a 9-fold pattern of 81 insights presented
as phrases, 2003). In the case of the T'ai Hsüan Ching,
each of the 81 insights ("Heads"), has a title and is explicated
through 9 very short philosophical verses (or "Appraisals", known
as Tsan),
typically presented in allegorical form -- and totalling 729 (as analyzed
by Walters) or 731 (as analyzed by Nyland and Sivin). Although not immediately
relevant to the following experiment, it is appropriate to note that each
of the 81 insights is linked to one of the 64 hexagrams of the I Ching (with
some duplication, of course) to evoke the old meanings and associations.

For Nylan and Sivin:

The philosophic interest of Yang's Head texts lies in their intricate, nuanced picture of a grand cycle of change, his recognition of complexity within regular order. Conversely, his Appraisals are remarkable because, through highly figurative language and the interplay of cycles within cycles, Yang suggests regular patterns emerging from the inexhaustible variety and ambiguity of moral circumstances.

The sequence of 81 insights of the Tao Te Ching are
however typically not clustered in any way. In the previous
experiment, the conventional order was used to cluster the 81 insights
into a 9 x 9 table to derive groups of 9 insights by row and, separately,
by column. The question was whether a magic
square clustering, much favoured
in that period in China, would enable new insights to be elicited from the
resultant pattern. It is important to note that the mathematical properties
of magic squares continue to be of great interest to mathematicians -- but
very little attention is paid to their potential role in ordering systems
of concepts. In that period however the 3 x 3 Lo
Shu magic square was essential
to the ordering of the most fundamental Chinese insights.

The 81 insights of the T'ai Hsüan Ching were
much more closely associated with magic square orderings than the Tao
Te Ching. The following experiment is based on the work of Derek Walters
(The T'ai Hsüan Ching: the hidden classic -- a lost companion of
the I Ching, 1983, subsequently titled The Alternative I Ching,
1987) who reconstructed and translated it. Walters notes the relationship
of the order of the T'ai Hsüan Ching to the arrangement of
the classic Magic Square of Master Tsan -- using the first modern numbering
system of the Han dynasty (a base ten system like that of the Romans) [more].
Walters explores the use of magic squares as a means of ordering the sets
of philosophical verses (tsan) clarifying each of the 81 insights.

Prior to Nylan, Walters is emphatic at the contrast between
the T'ai Hsüan Ching and the earlier I Ching it was
designed to improve upon. He notes that in the I Ching:

the texts of the hexagrams are an assembly of heterogeneous
phrases which range from magic formulas to complete poems, often repeated
in other hexagrams (although he later notes that the texts of the T'ai
Hsüan Ching are "a curious mixture of the deeply philosophical
and the domestically banal; the romantically poetic and the tritely mundane;
the clear and the abstruse")

the unexplained random order contrasts with the logical
sequence of the T'ai Hsüan Ching based on a unifying philosophy

The principal difference for Walters is that:

the T'ai Hsüan Ching holds that there are
three forces at work in nature; two of these Yin and Yang, represent the
positive and negative fluxes of electro-magnetism; but there is also a
third force which accounts for the creation of the truly novel. With Yin
and Yang [alone] there is nothing new under the Sun. While everything can
be classified as belonging to Yin and Yang, the dual philosophy can only
account for what exists already; no matter what resultant products or ideas
are spawned by the action of Yin and Yang, there is no entirely new element
created...But new ideas, and new species, only arise from the third creative
force... The third force, according to the T'ai Hsüan Ching,
is the Jen (Mankind) force... the philosophy of the triad is totally at
variance with the duality of the I Ching, and yet, paradoxically as though
it might seem the T'ai Hsüan Ching is a work of tremendous
originality (pp 8-9).

Like Walters, Nylan claims special advantages for the 'tri-partite'
division of the Tai Hsuan Ching. Walters had argued that the Tai
Hsuan Ching accorded a more complete, active role to 'man' --
as against the Yi-Chings
allegedly 'fixed' dualistic system. All of this shows a poor grasp of what
the Yi Ching (and Tai Hsuan Ching) actually teach. The
Tso chuan section of the Yi-Ching stresses that 'Heaven,
Earth and Man' are what comprise the Tao. Every trigram (and hexagram)
in the Yi Ching reflects this threefold unity ('the 'three
powers' or san-tsai) - and
the Yi-Ching makes this clear on every count. Human 'agency' is
therefore vital to the Yi-Ching. It was not a 'new' idea with
the Tai
Hsuan Ching.
The antiquity of this intuition is evident in the formation of the Chinese
script, the old Ku-wen forms, giving the character for 'king' or 'kingship'
as a representation of the san-tsai or 'three powers' -- linked
by a vertical stroke, anciently, the kingly-priest in whom the san-tsai
were united or focused. It is a basically a 'trigram' -- crossed by a vertical
line. Like Walters, Nylan makes some rather bold claims for the Tai
Hsuan Ching. But trying to place it in 'competition' against the Yi-Ching --
is naive. [more]

Part of the challenge of this experiment is to render comprehensible
a dynamic framework through which such differences of perspective, so typical
of academic dialogue, are understood as intrinsic to psychosocial dynamics
rather than in some way external to them. The widespread emphasis on correct
and incorrect views too readily reinforces the style of binary discourse
that justifies bloody conflicts -- as common today as at the time at which
such works originated -- and a reason for their elaboration.

Experiment

This following very simple experiment associates the titles
(as translated by Walters in one or two words) of the 81 insights of the T'ai
Hsüan Ching to the single-phrase presentation of the Tao
Te Ching -- in the tabular form explored previously. A choice was
however made to convert the terms chosen by Walters into gerund form (of
a synonym, if necessary), where this was not already the case. For example,
Walters has #1 as The Center, converted here to Centering;
he has #2 as Surrounding, not converted here, etc. The reason
for this is an interest here in the dynamic associated
with the insight rather than reinforcing any static sense
of a description. This adaptation may indeed be misleading given that the
version presented by Walters typically offers only a metaphoric allusion
to the sense of the insight.

Note that, following Walters, Michael Nylan (The Canon
of Supreme Mystery by Yang Hsiung. 1993) produced a new translation
of the T'ai Hsüan Ching using other English title variants
in some cases. The representation of the Tai Xuan Jing tetragram symbols
according to the web Unicode 5.0 standard (range
1D300-1D35F) uses another set of titles but specifically notes that these are not correct translations of the usual Chinese terminology.

The question raised by this experiment is whether the
titles (from the T'ai Hsüan Ching) and the phrase (from the Tao Te Ching) then have
any relationship -- whether inherently meaningful, intuitive, or aesthetically
suggestive --
despite the complex pathways through which the English texts were derived,
and the criticism that could be validly made of this approach. Minimally
however it permits an inspection of the juxtaposed elements from quite
different sources. It is possible that any relationship that may exist
is necessarily to be understood as a paradoxical challenge to comprehension
like a Zen koan (gōng-àn in
Chinese).

The configurations of insights in the following tables point
to the possibility that, like stringed instruments, they in all probability
each require a form of semantic or memetic "tuning" to be able
to communicate the interplay of insights whose totality is named by the author
as "mystery".
To what extent do the interlocking numbers in any magic square patterning
enable such comprehension?

The magic square configuration, of significance at that time,
may help to determine whether the relatiuonship between the insights of the
Tao Te Ching and the T'ai Hsüan Ching is very significant
or fortuitous at best.

Experimental presentation

Table 1: The basis for the following
table of the 81 insights of the Tao Te Ching is discussed in a separate
commentary. The rows of the table provide 9 groups in terms of the conventional
ordering in the Tao Te Ching. The columns of the table provide
9 different groups in terms of the alternative
ordering represented by those columns. The title from the Tai
Hsüan Ching (or adapted therefrom) has been added before each phrase, in italics,
respecting the number order.

80: Attending: Enjoying the
freedom of movement in relation to what is to hand

81: Nourishing: Doing without
outdoing

Magic square presentation

Table 2: The basis for the following table of the
81 insights of the Tao Te Ching is discussed in a separate
commentary. It is an experiment in the organization of these insights
into clusters. The table is made up of 9 nested tables (each of 9 cells). Each
nested table corresponds to one of the rows from Table
1above -- each row above being transformed into a nested
table of 3x3 cells below. Note that the insight numbers in each row total
to 369, as do the insight numbers in each column.

71: Stopping:
Knowing without knowing

64: Sinking:
Attending to what may have been neglected in the achievement of undertakings

69: Lacking:
Yielding to antagonism

8: Opposing:
Easing forward, going wherever, without competition

1:Centering:
Journeying through unnaming the myriad patterns of the past

6: Suffering:
Completing

53: Perpetuating:
Ensuring modesty

46:Extending:
Knowing that enough is enough

51: Ruling:
Nurturing life according to natural processes

66: Departing:
Following rather than leading

68: Dimming:
Avoiding competition

70: Depriving:
Being obscure

3: Slowing:
Cultivating non-engagement

5: Lessening:
Engendering through complementarity

7: Ascending:
Enduring

48: Fitting:
Unlearning

50: Doddering:
Living in recognition that this implies dying

52: Placing:
Understanding insignificant beginnings

67: Obscuring:
Leading the mightiest by not presuming to do so

72: Persevering:
Fearing the dangers of acting inappropriately

65: Questioning:
Being in ignorance of appropriate action

4: Obstructing:
A Barrier: Having been there; having done that

9: Penetrating:
Avoiding excess

2:Surrounding:
Engaging without engaging

49: Escaping:
Enminding the world to see the ordinary through the eyes of children

54: Ensuring:
Ensuring that rules for oneself are consistent with those for the world

47: Comprehending:
Understanding the truth and opportunity of the moment

31: Pretending:
Using weapons, when there is no choice, with a calm, still mind

36: Enclosing:
Prevailing through weakness

29: Severing:
Doing "nothing"
to the world

76: Distressing:
Bending in response to pressure

81: Nourishing:
Doing without outdoing

74: Willing:
Avoiding the presumptuousness of usurping the judgement on others

13: Increasing:
Governing others appropriately

18: Waiting:
Failing to exalt merit

11: Mistaking:
Benefiting from what is not

4:96

9:231

2:42

As a further experiment in organization, the insights were clustered according
to the mathematical principle of the magic square (see Table
2). The structure of Table
2 is best understood by considering the first row of 9 insights (1 to 9)
in Table 1.
These 9 appear as the central nested table in the top row of 3 nested
tables in Table
2. The 9 in that nested table are however presented in an order based on
the structure of what is known in mathematics as a magic
square -- -- namely the numbers of the insights (of the conventional
ordering in the Tao Te Ching), whatever the direction of addition,
whether vertically (8+3+4; 1+5+9; 6+7+2), horizontally (8+1+6; 3+5+7; 4+9+2),
or diagonally (8+5+2; 4+5+6), total in each case to 15 (as indicated there as
1:15). Similarly if the numbers of each row are multiplied (8x1x6; 3x5x7; 4x9x2)
they together total to 225 -- as do those of the columns (8x3x4; 1x5x9; 6x7x2).

In such a square the numbers of the first 9 insights (1 to 9) (of the conventional
ordering in the Tao Te Ching), whatever the direction of addition,
whether vertically (8+3+4; 1+5+9; 6+7+2), horizontally (8+1+6; 3+5+7; 4+9+2),
or diagonally (8+5+2; 4+5+6), total in each case to 15 (as indicated there as
1:15). Similarly if the numbers of each row are multiplied (8x1x6; 3x5x7; 4x9x2)
they together total to 225 -- as do those of the columns (8x3x4; 1x5x9; 6x7x2).

This is an adaptation of the Lo-Shu
order known in classical China. In the table as a whole, the 9 nested tables
have been positioned in a manner corresponding to this same order. Thus
the first row of nested tables in Table
2 (above) groups the contents of rows 8, 1 and 6 respectively from Table
1 (namely rows marked there as VIII, I, and VI), the second groups 3, 5
and 7, with the third grouping 4, 9 and 2. The principle of the magic square
is discussed elsewhere
(notably by Alan Grogono), together
with its long history dating back to 2800 BC [more
| more
| more | more].

The Lo Shu is the only magic square of order 3. Namely there is just
one 3x3 magic square -- although with rotations and reflections, there are eight
variations of what is essentially the same square. An associative
magic square of order n is one for which every pair of numbers symmetrically
opposite the center sum to n2+1. The Lo Shu square is associative
-- but is not a panmagic
square for which all the diagonals --including the broken diagonals obtained
by "wrapping around" the edges -- total like the rows and columns.

Just as the magic square total for the first 3x3 nested table is 15 (indicated
above in Table
2 as 1:15), each other 3x3 nested table gives rise to its own total (indicated
beneath it, eg 4:96, 9:231, and 2:42). The 9 such totals from each nested table
also constitute a magic square -- with a total figure of 369. As might be expected,
if the table as a whole is treated as a 9x9 magic square, the total is also
369.

Interesting patterns
can be generated from magic squares when the numbers of the squares are replaced
by symmetric symbols.

Pan-magic square presentation

Mathematically a "continuous" ("pan-magic", pan-diagonal,
Nasik or Jaina) square has the additional property that even the broken
diagonals add to the same total as those of the magic square. It was long
supposed that a 9x9 pan-magic square did not exist, but one such based on the
81 numbers 0 to 80 is reported by Alan Grogono [more].
He explains this early belief as probably due to the absence of any obvious
pattern to use to create a regular 9x9 square. Constructing a square by expanding
a 3x3 square indeed produces a magic square as in Table
2 but not a pan-magic one. In addition, amongst odd-order pan-magic squares,
most interest has been focused on the regular prime number squares. These lent
themselves to analysis more readily and to calculation of the number of regular
pan-magic squares which could be constructed with an underlying pattern.

Grogono argues that the analysis (and construction) of magic squares is more
logical, and the results make more sense, when the smallest number is 0 -- instead
of 1. This would imply that a 9x9 square of the Tao Te Ching insights
should run from 0 to 80 instead of from 1 to 81. This would not affect the pattern
of Table
2, provided that the rows from which it was derived in Table
1 were then renumbered from 0 to 8 (instead of from I to IX).

Of further interest, however, is to use the 9x9 pan-magic square order discovered
by Grogono to redistribute the 81 insights. There is an interesting clue
to the relevance of renumbering the first insight from 1 to 0 -- in the text
of that first insight itself.

Given the properties of the pan-magic square, in this case the row containing
0 (the insight traditionally numbered 1) in his case was shifted to the central
position (and checked in the online facility he provides to ensure that it remained
a pan-magic square). This gives the following (Table
3a) from which the ordering in Table
3b was then produced -- retaining the numbering of the insights in Table
1 (namely 0 in Table 3a is 1 in Table 3b, in order to correspond to
Table 1).

In 1999 Dan
Washburn made the point that "The vastu-purusha-mandala is a square
of 81 subsquares with 9 subsquares on each side. Take a Lo Shu magic
sqaure of 3 and place a Lo Shu magic square of 3 in each of its 9 subsquares
and you have a 9 x 9 square of 81 subsquares. So the vastu-purusha-mandala is
the Lo Shu square squared, or seen in more detail." According to Vini Nathan
(Vastu Purusha Mandala:
Beyond Building Codes, Nexus Network Journal, vol. 4, no. 3,
Summer 2002), The Vastu purusha mandala has been defined as "a collection of
rules which attempt to facilitate the translation of theological concepts into
architectural form." This law of proportions and rhythmic ordering of elements
not only found full expression in temples, but extended to residential and urban
planning as well. He argues that the influence of the Vastu purusha mandala
extended beyond building activity to encompass the cultural milieu as well.

Since the ternary numbers go from 0 to 80 (as indicated by Grogono in Table
3a) instead of from 1 to 81, their equivalent in the Tai Hsuan Ching is
obtained by adding "plus 1" to the numbers in Table 3a, conserving
the order.

Note that the insight numbers in each row now total to 369 (as in Table
2, and in contrast to the 360 of Table
3a), as do the insight numbers in each column). In addition the total of
the insight numbers in any 3x3 nested square (even across highlighting)
also total to 369 -- whereas those of the 3x3 nested squares (even those highlighted)
in Table
2 are not equal (although those of the central 3x3 square only do
indeed total to 369). Note that the difference of 9 between 360 and 369 derives
from the difference in insight numbering from 0-80 against 1-81 (giving a difference
of 9, whether in row or column totals).

37:Encycling: Self-organizing
of myriad things

52: Placing: Understanding
insignificant beginnings

31: Pretending:
Using weapons, when there is no choice, with a calm, still mind

66: Departing:
Following rather than leading

81: Nourishing:
Doing without outdoing

60: Storing: Allowing
potentially disruptive forces to have their place

The sequence of numbers in the Tai Hsuan Ching is conventionally arranged
into three groups of three called T'ien (1-27), Jen (28-54)
and Ti (55-81)
as in the table below (which is not a magic square arrangement as explored here).

Ti
(yin)

Jen
(yin-yang)

T'ien
(yang)

73

64

55

46

37

28

19

10

1

74

65

56

47

38

29

20

11

2

75

66

57

48

39

30

21

12

3

76

67

58

49

40

31

22

13

4

77

68

59

50

41

32

23

14

5

78

69

60

51

42

33

24

15

6

79

70

61

52

43

34

25

16

7

80

71

62

53

44

35

26

17

8

81

72

63

54

45

36

27

18

9

This ternary number (+1) arrangement, according to Tony Smith, is similar to
the Fu
Xi binary
number arrangement of the I Ching (although this is not a magic
square arrangement). The positions in Table 3b can now be coloured
in Table 3c (below) to highlight these three groups.

37:Encycling: Self-organizing
of myriad things

52: Placing: Understanding
insignificant beginnings

31: Pretending:
Using weapons, when there is no choice, with a calm, still mind

66: Departing:
Following rather than leading

81: Nourishing:
Doing without outdoing

60: Storing: Allowing
potentially disruptive forces to have their place

74: Willing: Avoiding
the presumptuousness of usurping the judgement on others

7: Ascending:
Enduring

13: Increasing:
Governing others appropriately

19:Conforming:
Being untroubled through needing little and wanting less

36: Enclosing:
Prevailing through weakness

42: Greeting:
Losing as the key to the cycle of winning and losing

48: Fitting: Unlearning

12: Rejuvenating:
Sensing the inner

27:Transacting:
Educating the challenged as the inspiration of the wise

6: Suffering:
Completing

38: Abounding:
Abiding in letting go and doing nothing

53: Perpetuating:
Ensuring modesty

32: Multiplying:
Knowing when to cease making essential distinctions

64: Sinking: Attending
to what may have been neglected in the achievement of undertakings

79: Troubling:
Fulfilling obligations

58: Re-uniting:
Bumbling on without forcing

The property of pan-magic squares whereby there is a continuity of the "magic"
property even along the broken diagonals may be consistent with the phenomenon
expressed poetically in the much-quoted stanza of T
S Eliot (in Little
Gidding, 1942):

We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time.

Bimagic squares

Mathematically a magic square is bimagic (or 2-multimagic) if it remains "magic"
after each of its numbers have been squared -- a bimagic square thus has the
additional property that if each number in the square is multiplied by itself
(squared, or raised to the second power) the resulting row, column, and diagonal
sums are also magic. Bimagic squares are a subset of the class of multimagic
squares; it is believed that no bimagic squares of order less than 8 exists
(Benson and Jacoby 1976). The original 3x3 Lo Shu square is far from
being bimagic, since the sums of the squared numbers (of the rows or columns)
vary between 77 and 107. The discoverer of the first bimagic square, G. Pfeffermann
later published in Les
Tablettes du Chercheur (15 July 1891) the first 9th-order bimagic square.
In the case of the examples of bimagic squares based on 9x9 in Table
4 (below), the rows and columns sum to 369 as before. But if each number
is squared, the sum is then 20,049.

Most-perfect magic squares

A special type of pan-diagonal magic square is characterized as most-perfect
[more]. An
example of a 12x12 most-perfect magic square is provided by Ian Stewart [more].
The numbers in every 2x2 square sum to 286. More generally every 2 x 2 block
of cells (including wrap-around) sum to 2T (where T= n2 + 1). Any
pair of integers distant ** along a diagonal sum to T.

Magic cubes

There are extensive resources on magic cubes and hypercubes [notably Harvey
Heinz and Marián
Trenkler] that may offer even more powerful ways of organizing the 81 insights.
A magic cube is a three-dimensional version of the magic square in which the
rows, columns, pillars (or "files"), and four space diagonals each sum to a
single number known as the magic
constant. If the cross section diagonals also sum to that constant, the
magic cube is called a perfect magic cube; if they do not, the cube is called
a semiperfect magic cube, or sometimes an Andrews cube (Gardner 1988). A pandiagonal
cube is a perfect or semiperfect magic cube which is magic not only along the
main space diagonals, but also on the broken space diagonals [more].
In a panmagic square, in addition to the main diagonals, the broken diagonals
also sum to the magic constant.

Simple: Containing no, or less then 3m, orthogonal magic squares.
Only the rows, columns, pillars and triagonals are required to sum correctly
for a simple magic cube.

Pantriagonal: All pantriagonals must sum correctly. There may be
some simple and/or pandiagonal magic squares, but not enough to satisfy any
other classifications.

Diagonal: All 3m planar arrays must be 'simple' magic squares.

Pandiagonal: All 3m planar arrays must be 'pandiagonal' magic squares.
The 6 oblique squares are always magic. One of them may be pandiagonal magic.

Perfect: All 3m planar arrays must be 'pandiagonal' magic squares.
In addition, all pantriagonals must sum correctly. These two conditions combine
to provide another 6m pandiagonal magic squares.

Heinz notes that a magic cube is called normal if it consists of the
numbers 1 to m3 (or 0 to m3 - 1). A magic cube is called
associated if all pairs of two numbers diametrically equidistant from
the center of the cube equal the sum of the first and last number in the series.
If the associated cube (or other dimension of hypercube) is an odd order, then
the center of the cube is a cell containing one half the sum of the first and
last number in the series.

Heinz provides a generalized definition as follows: A hypercube of dimension
n is perfect if all pan-n-agonals sum correctly, and all lower dimension hypercubes
contained in it are perfect! He also provides spreadsheets for testing them.
Heinz has collaborated with J. R. Hendricks to produce a A Unified Classification
system for Magic Cubes (Journal of Recreational Mathematics, 2002).

To construct the Tai Hsuan Ching, consider the Magic Square sequence
as a line 3 8 4 9 5 1 6 2 7 with central 5 and opposite pairs at equal distances.
If you try to make that, or a multiple of it, into a 9x9 Magic Square whose
central number is the central number 41 of 9x9 = 81 = 40+1+40, you will fail
because 41 is not a multiple of 5.

However, since 365 = 5x73 is the central number of 729 = 364+1+364, you
can make a 9x9x9 Magic Cube with 9x9x9 = 729 entries, each 9x9 square of which
is a Magic Square. The Magic Cube of the Tai Hsaun Ching gives the
same sum for all lines parallel to an edge, and for all diagonals containing
the central entry. The central number of the Magic Cube, 365....

The total number for each line is 3,285 = 219 x 15. The total of all numbers
is 266,085 = 5,913 x 45.

Since 729 is the smallest odd number greater than 1 that is both a cubic
number and a square number, the 729 entries of the 9x9x9 Magic Cube with central
entry 365 can be rearranged to form a 27x27 Magic Square with 729 entries
and central entry 365. 27 = 3x3x3 = 13+1+13 is a cubic number with central
number 14, and there is a 3x3x3 Magic Cube with central entry 14 (14 is the
dimension of the exceptional Lie algebra G2) and sum 42...

Magic hypercubes

A magic tesseract
is a four-dimensional generalization of the two-dimensional magic square and
the three-dimensional magic cube. Harvey Heinz defines a 4-dimensional hypercube
(or tesseract) as perfect if all pan-quadragonals are correct, and all the
magic squares and magic cubes within it are perfect. This means that the
magic squares are all pandiagonal and the magic cubes are all pantriagonal
and pandiagonal. There are 40m2 lines that sum correctly. They
are m3 rows,
m3 columns, m3 pillars, m3 files, 8m3
quadragonals, 16m3 triagonals, and 12m3 diagonals. Furthermore,
a magic hypercube of any dimension n is perfect if all pan-n-agonals sum correctly,
and all lower dimension hypercubes contained in it are perfect!

Conclusion

Natural appropriateness: Yang Hsiung is understood to have
shared the vision of the I
Ching of
an order that united the cosmos, the sphere of action, and the individual.
The concern of all such works, notably the Tao Te Ching, might be
said to be with appropriateness in terms of the patterns and dynamics of nature
-- the Way of Nature, exemplified by the insights of the Tao Te Ching.
Any explorations of such matters merit careful consideration in the light of
contemporary challenges of global governance of an endangered planet -- exemplified
by the paradox of "sustainable development".

In the Way, which was also Yang Hsiung's Mystery, science and ethics were one. This Han vision was not a reduction of science to subjectivity. Nature and human nature constitute a single order.... "Hsuan" carries a range of meaning from "black" to "darkness" to "hidden" to "mystery." Its overtones are stillness, solitude, isolation, nondifferentiation, and inaccessibility by purely rational processes. In Chinese thought the ideas at the philosophical end of this range bear no unpleasant connotations. They express that aspect of experience that can be known only by quiet and deep contemplation, or by illumination. Yang Hsiung uses hsuan in his book's title and throughout to mean the profound darkness, silence, ambiguity, and indefiniteness out of which creation comes. In cosmogony it is the undifferentiated state out of which yin-yang and eventually the myriad phenomena separate. In Nature as humans experience it, it is the latency out of which individual things are spontaneously born, and out of which events shape themselves. In the sage--that is, the human being as he should be, as the student of the Mystery is striving to be--it is the spiritual inwardness that precedes conscious decision and action and spontaneously accords them with natural process.

Complementarity: For its author, Yang Hsiung, the intent was to evoke a sense
of the inseparability and complementarity between mystery and rational pattern.
Again for Nylan and Sevin:

Despite the superficiality of this essay, we have adduced more than enough evidence to illustrate Yang's aim in writing the Mystery: to instigate and guide the personal striving for integrity that is the only possible basis for a sound polity. This virtue is more than a matter of moral and psychic integration; it involves union with the Way of Nature and its Mystery. As the inquirer aligns his own decisions and actions with the cosmic course, he is able to promote harmony in every external sphere, renovating society on the pattern of the Tao.

Degree of relationship: With respect to the significance
of the pattern of relationships between the T'ai Hsüan Ching and
the Tao Te
Ching --
both pointing to a subtle understanding -- the tables above could
be usefully seen as raising questions as to the nature or degree of that relationship,
or of any ability to recognize it. It is of course highly unsatisfactory to
assess such a relationship through a matching of single-word insight titles
from the T'ai
Hsüan
Ching to
succinct representations of insights from the Tao Te Ching. Each such "insight" is
a memetic complex at the nexus of the pattern of associations it evokes. It
is severely distorted by reducing its dimensionality in this way through the
use of a somewhat arbitrary choice of term in English.

Might there therefore be an analogous phenomenon in the knowledge universe
whereby even the most seemingly disparate concepts could be linked through
a short chain of semantically relevant associations?

As with a song or poem, it is the
recurring elements and associations that ensure the integrity and memorability
of the whole -- arguments developed in an associated proposal (A
Singable Earth Charter, EU Constitution or Global Ethic?, 2006).
A sense of the challenge, and willingness to address it, may be seen in current
widespread enthusiam for sudoku --
which have their origin in magic squares. The objective is to achieve interlocking
of symbols (of any kind) in an array -- an intellectually "satisfying" design
"fit". The 81 memetic complexes of the T'ai
Hsüan Ching and the Tao Te Ching pose the challenge of
how to identify, understand, remember and communicate an appropriately interlocked
pattern -- of relevance to an appropriate lifestyle and style of governance.
As with sudoku, they point to the possibility of more challenging
complex patterns of interconnectedness (including bimagic squares, cubes
and hypercubes).

Individual engagement: The additional dimension introduced by the preoccupations
of the two works linked here, notably through the 9x9 pan-magic square presentation
(centered on #1: Centering), is the sense of a central focus for order
with which the individual may fruitfully identify. This can be contrasted with
conventional systems diagrams that have no such centre, singificantly failing
to design in the individual observer, as a focus for action within the system.
The closest to this is ironically the arrow on a subway may indicating the
user's current location. However the "centering" in the pan-magic
square is an extremely subtle notion that, paradoxically and provocatively,
calls into question the naming and condition of the centre itself -- as often
quoted from the Tao
Te Ching in many variations :

The way that can be told is not the common way
The name that can be named is
not the common name
What has no name is the beginning of heaven and earth
What has a name is the mother of the myriad creatures
Those without desires
contemplate its secrets
Those who have desires contemplate its periphery
These two emerge together, but differ in name
Being together, they are called
Mystery
Mystery upon mystery
Gateway to the myriad secrets.
[Nylan amd Sirvin]

Readers of any systems diagram, like the magic squares above, are therefore
challenged as much to ask "who they think they are" and "where
they think they are going", rather than simply to expect a simple answer
that in no way challenges their sense of identity or the validity of their
quest. They offer a puzzle that merits being taken seriously.

Focus of cognitive engagement: It is from such a perspective that it is worth
recalling that magic squares were traditionally placed at the centre of mandalas as
a focus for meditation. These mandalas might then be viewed as a form of cognitive "gearbox" through
which "power is transmitted to an output device, normally
rotary in form, and generally at a reduced rate of angular speed but at a higher
motive torque". The "gears" presented in such mandalas typically
involve concentric sets of (3, 4, 5, 6, 7, 8, 9, 10 or 12) archetypal symbols
as conduits for particular modes of understanding. From such a perspective,
the challenge of appropriateness for an individual lifestyle, and for collective
governance, is to identify such "gears" and
to ensure that they intermesh fruitfully.

The Elemental Changes -- the ancient Chinese companion
to the I Ching: the T'ai Hsuan Ching of Master Yang Hsiung. Albany,
State University of New York Press, 1993 (abridged version of The Canon
of Supreme Mystery)